Joint Entrance Examination

Graduate Aptitude Test in Engineering

Geomatics Engineering Or Surveying

Engineering Mechanics

Hydrology

Transportation Engineering

Strength of Materials Or Solid Mechanics

Reinforced Cement Concrete

Steel Structures

Irrigation

Environmental Engineering

Engineering Mathematics

Structural Analysis

Geotechnical Engineering

Fluid Mechanics and Hydraulic Machines

General Aptitude

1

Let

A = $$\left\{ {\theta \in \left( { - {\pi \over 2},\pi } \right):{{3 + 2i\sin \theta } \over {1 - 2i\sin \theta }}is\,purely\,imaginary} \right\}$$

. Then the sum of the elements in A is :

A = $$\left\{ {\theta \in \left( { - {\pi \over 2},\pi } \right):{{3 + 2i\sin \theta } \over {1 - 2i\sin \theta }}is\,purely\,imaginary} \right\}$$

. Then the sum of the elements in A is :

A

$${5\pi \over 6}$$

B

$$\pi $$

C

$${3\pi \over 4}$$

D

$${{2\pi } \over 3}$$

Given complex number,

$${{3 + 2i\sin \theta } \over {1 - 2i\sin \theta }}$$

$$ = {{\left( {3 + 2i\sin \theta } \right)\left( {1 + 2i\sin \theta } \right)} \over {1 + 4{{\sin }^2}\theta }}$$

$$ = {{3 + 6i\sin \theta + 2i\sin \theta - 4{{\sin }^2}\theta } \over {1 + 4{{\sin }^2}\theta }}$$

$$ = {{\left( {3 - 4{{\sin }^2}\theta } \right) + i\left( {8\sin \theta } \right)} \over {1 + 4{{\sin }^2}\theta }}$$

As complex number is purely imaginary, So real part of this complex number is zero.

$$ \therefore $$ $${{3 - 4{{\sin }^2}\theta } \over {1 + 4{{\sin }^2}\theta }}$$ = 0

$$ \Rightarrow $$ $$3 - 4{\sin ^2}\theta = 0$$

$$ \Rightarrow $$ $$\sin \theta = \pm {{\sqrt 3 } \over 2}$$

as $$\theta $$ $$ \in $$ $$\left( { - {\pi \over 2},\pi } \right)$$

$$ \therefore $$ $$\theta $$ $$=$$ $$-$$ $${\pi \over 3},{\pi \over 3},{{2\pi } \over 3}$$

$$ \therefore $$ Sum of those values of A is

$$ = - {\pi \over 3} + {\pi \over 3} + {{2\pi } \over 3}$$

$$ = {{2\pi } \over 3}$$

$${{3 + 2i\sin \theta } \over {1 - 2i\sin \theta }}$$

$$ = {{\left( {3 + 2i\sin \theta } \right)\left( {1 + 2i\sin \theta } \right)} \over {1 + 4{{\sin }^2}\theta }}$$

$$ = {{3 + 6i\sin \theta + 2i\sin \theta - 4{{\sin }^2}\theta } \over {1 + 4{{\sin }^2}\theta }}$$

$$ = {{\left( {3 - 4{{\sin }^2}\theta } \right) + i\left( {8\sin \theta } \right)} \over {1 + 4{{\sin }^2}\theta }}$$

As complex number is purely imaginary, So real part of this complex number is zero.

$$ \therefore $$ $${{3 - 4{{\sin }^2}\theta } \over {1 + 4{{\sin }^2}\theta }}$$ = 0

$$ \Rightarrow $$ $$3 - 4{\sin ^2}\theta = 0$$

$$ \Rightarrow $$ $$\sin \theta = \pm {{\sqrt 3 } \over 2}$$

as $$\theta $$ $$ \in $$ $$\left( { - {\pi \over 2},\pi } \right)$$

$$ \therefore $$ $$\theta $$ $$=$$ $$-$$ $${\pi \over 3},{\pi \over 3},{{2\pi } \over 3}$$

$$ \therefore $$ Sum of those values of A is

$$ = - {\pi \over 3} + {\pi \over 3} + {{2\pi } \over 3}$$

$$ = {{2\pi } \over 3}$$

2

Let z_{0} be a root of the quadratic equation, x^{2} + x + 1 = 0, If z = 3 + 6iz$$_0^{81}$$ $$-$$ 3iz$$_0^{93}$$, then arg z is equal to :

A

$${\pi \over 4}$$

B

$${\pi \over 6}$$

C

$${\pi \over 3}$$

D

0

1 + x + x^{2} = 0

x = $${{ - 1 \pm \sqrt {1 - 4} } \over 2} = {{ - 1 \pm i\sqrt 3 } \over 2}$$

z_{0} = w, w^{2}

Now

z = 3 + 6iz$$_0^{81}$$ $$-$$ 3iz$$_0^{93}$$

z = 3 + 6iw^{81} $$-$$ 3iw^{93} (w^{93} = w^{81} = 1)

$$ \Rightarrow $$ z = 3 + 3i

then arg(z) = tan^{$$-$$1}$$\left( {{3 \over 3}} \right)$$ = tan^{$$-$$1} (1) = $${\pi \over 4}$$

x = $${{ - 1 \pm \sqrt {1 - 4} } \over 2} = {{ - 1 \pm i\sqrt 3 } \over 2}$$

z

Now

z = 3 + 6iz$$_0^{81}$$ $$-$$ 3iz$$_0^{93}$$

z = 3 + 6iw

$$ \Rightarrow $$ z = 3 + 3i

then arg(z) = tan

3

Let z_{1} and z_{2} be any two non-zero complex numbers such that $$3\left| {{z_1}} \right| = 4\left| {{z_2}} \right|.$$ If $$z = {{3{z_1}} \over {2{z_2}}} + {{2{z_2}} \over {3{z_1}}}$$ then -

A

$${\rm I}m\left( z \right) = 0$$

B

$$\left| z \right| = \sqrt {{17 \over 2}} $$

C

$$\left| z \right| =$$ $${1 \over 2}\sqrt {9 + 16{{\cos }^2}\theta } $$

D

Re(z) $$=$$ 0

Given, $$3\left| {{z_1}} \right| = 4\left| {{z_2}} \right|$$

$$ \Rightarrow $$ $${{\left| {{z_1}} \right|} \over {\left| {{z_2}} \right|}} = {4 \over 3}$$

$$ \Rightarrow $$ $${{\left| {3{z_1}} \right|} \over {\left| {2{z_2}} \right|}} = {4 \over 3} \times {3 \over 2} = 2$$

As we know, for any compled number

$${{3{z_1}} \over {2{z_2}}} = {{\left| {3{z_1}} \right|} \over {\left| {2{z_2}} \right|}}$$(cos$$\theta $$ + i sin$$\theta $$)

= 2(cos$$\theta $$ + i sin$$\theta $$)

$$ \therefore $$ $${{2{z_2}} \over {3{z_1}}}$$ = $${1 \over {2\left( {\cos \theta + i\sin \theta } \right)}}$$

= $${1 \over {2\left( {\cos \theta + i\sin \theta } \right)}} \times {{\left( {\cos \theta - i\sin \theta } \right)} \over {\left( {\cos \theta - i\sin \theta } \right)}}$$

= $${\left( {{1 \over 2}\cos \theta - {i \over 2}\sin \theta } \right)}$$

Now, given

$$z = {{3{z_1}} \over {2{z_2}}} + {{2{z_2}} \over {3{z_1}}}$$

= 2(cos$$\theta $$ + i sin$$\theta $$) + $${\left( {{1 \over 2}\cos \theta - {i \over 2}\sin \theta } \right)}$$

= $${{5 \over 2}\cos \theta + {3 \over 2}i\sin \theta }$$

So, |z| = $$\sqrt {{{25} \over 4}{{\cos }^2}\theta + {9 \over 4}{{\sin }^2}\theta } $$

= $${1 \over 2}\sqrt {9 + 16{{\cos }^2}\theta } $$

z is neither purely real nor purely imaginary and |z| depends on $$\theta $$.

$$ \Rightarrow $$ $${{\left| {{z_1}} \right|} \over {\left| {{z_2}} \right|}} = {4 \over 3}$$

$$ \Rightarrow $$ $${{\left| {3{z_1}} \right|} \over {\left| {2{z_2}} \right|}} = {4 \over 3} \times {3 \over 2} = 2$$

As we know, for any compled number

$${{3{z_1}} \over {2{z_2}}} = {{\left| {3{z_1}} \right|} \over {\left| {2{z_2}} \right|}}$$(cos$$\theta $$ + i sin$$\theta $$)

= 2(cos$$\theta $$ + i sin$$\theta $$)

$$ \therefore $$ $${{2{z_2}} \over {3{z_1}}}$$ = $${1 \over {2\left( {\cos \theta + i\sin \theta } \right)}}$$

= $${1 \over {2\left( {\cos \theta + i\sin \theta } \right)}} \times {{\left( {\cos \theta - i\sin \theta } \right)} \over {\left( {\cos \theta - i\sin \theta } \right)}}$$

= $${\left( {{1 \over 2}\cos \theta - {i \over 2}\sin \theta } \right)}$$

Now, given

$$z = {{3{z_1}} \over {2{z_2}}} + {{2{z_2}} \over {3{z_1}}}$$

= 2(cos$$\theta $$ + i sin$$\theta $$) + $${\left( {{1 \over 2}\cos \theta - {i \over 2}\sin \theta } \right)}$$

= $${{5 \over 2}\cos \theta + {3 \over 2}i\sin \theta }$$

So, |z| = $$\sqrt {{{25} \over 4}{{\cos }^2}\theta + {9 \over 4}{{\sin }^2}\theta } $$

= $${1 \over 2}\sqrt {9 + 16{{\cos }^2}\theta } $$

z is neither purely real nor purely imaginary and |z| depends on $$\theta $$.

4

Let $$z = {\left( {{{\sqrt 3 } \over 2} + {i \over 2}} \right)^5} + {\left( {{{\sqrt 3 } \over 2} - {i \over 2}} \right)^5}.$$ If R(z) and 1(z) respectively denote the real and imaginary parts of z, then -

A

R(z) = $$-$$
3

B

R(z) < 0 and I(z) > 0

C

I(z) = 0

D

R(z) > 0 and I(z) > 0

$$z = {\left( {{{\sqrt 3 + i} \over 2}} \right)^5} + {\left( {{{\sqrt 3 - i} \over 2}} \right)^5}$$

$$z = {\left( {{e^{i\pi /6}}} \right)^5} + {\left( {{e^{ - i\pi /6}}} \right)^5}$$

$$ = {e^{i5\pi /6}} + {e^{ - i5\pi /6}}$$

$$ = \cos {{5\pi } \over 6} + i{{\sin 5\pi } \over 6} + \cos \left( {{{ - 5\pi } \over 6}} \right) + i\sin \left( {{{ - 5\pi } \over 6}} \right)$$

$$ = 2\cos {{5\pi } \over 6} < 0$$

$${\rm I}(z) = 0$$ and $${\mathop{\rm Re}\nolimits} (z) < 0$$

$$z = {\left( {{e^{i\pi /6}}} \right)^5} + {\left( {{e^{ - i\pi /6}}} \right)^5}$$

$$ = {e^{i5\pi /6}} + {e^{ - i5\pi /6}}$$

$$ = \cos {{5\pi } \over 6} + i{{\sin 5\pi } \over 6} + \cos \left( {{{ - 5\pi } \over 6}} \right) + i\sin \left( {{{ - 5\pi } \over 6}} \right)$$

$$ = 2\cos {{5\pi } \over 6} < 0$$

$${\rm I}(z) = 0$$ and $${\mathop{\rm Re}\nolimits} (z) < 0$$

Number in Brackets after Paper Name Indicates No of Questions

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Trigonometric Functions & Equations *keyboard_arrow_right*

Properties of Triangle *keyboard_arrow_right*

Inverse Trigonometric Functions *keyboard_arrow_right*

Complex Numbers *keyboard_arrow_right*

Quadratic Equation and Inequalities *keyboard_arrow_right*

Permutations and Combinations *keyboard_arrow_right*

Mathematical Induction and Binomial Theorem *keyboard_arrow_right*

Sequences and Series *keyboard_arrow_right*

Matrices and Determinants *keyboard_arrow_right*

Vector Algebra and 3D Geometry *keyboard_arrow_right*

Probability *keyboard_arrow_right*

Statistics *keyboard_arrow_right*

Mathematical Reasoning *keyboard_arrow_right*

Functions *keyboard_arrow_right*

Limits, Continuity and Differentiability *keyboard_arrow_right*

Differentiation *keyboard_arrow_right*

Application of Derivatives *keyboard_arrow_right*

Indefinite Integrals *keyboard_arrow_right*

Definite Integrals and Applications of Integrals *keyboard_arrow_right*

Differential Equations *keyboard_arrow_right*

Straight Lines and Pair of Straight Lines *keyboard_arrow_right*

Circle *keyboard_arrow_right*

Conic Sections *keyboard_arrow_right*